1.1.1 Goals of this lecture

• Extend 2 x 2 contingency tables to larger tables
  • More variable categories: \(2 \times 3\) and larger
  • More variables: \(2 \times 2 \times 2\) tables
• Chi-square tests for these tables
  • Probing the tables
  • Residuals

2.1.1 \(2 \times 2\) tables and beyond…

• We’ve only looked at \(2 \times 2\) tables
  • Extend these tables in terms of rows or columns
    • Just more rows: \(I \times 2\)
    • Just more columns: \(2 \times J\)
    • More rows and more columns: \(I \times J\)

2.1.2 Whickam2 data with a twist

   Outcome Smoker Age AgeGroup Alive
1    Alive    Yes  23    18-64     1
2    Alive    Yes  18    18-64     1
3     Dead    Yes  71      65+     0
4    Alive     No  67      65+     1
5    Alive     No  64    18-64     1
6    Alive    Yes  38    18-64     1
7    Alive    Yes  45    18-64     1
8     Dead     No  76      65+     0
9    Alive     No  28    18-64     1
10   Alive     No  27    18-64     1

• Split the Age variable into 3 groups
  • 18 to 40
  • 41 to 64
  • 65+

2.1.3 Convert Age into 3 categories

Whickham2 <- Whickham2 %>%
    mutate(AgeGroup3 = ifelse(Age %in% 18:40, 1,
           ifelse(Age %in% 41:64, 2, 3)))
head(Whickham2)

  Outcome Smoker Age AgeGroup Alive AgeGroup3
1   Alive    Yes  23    18-64     1         1
2   Alive    Yes  18    18-64     1         1
3    Dead    Yes  71      65+     0         3
4   Alive     No  67      65+     1         3
5   Alive     No  64    18-64     1         2
6   Alive    Yes  38    18-64     1         1

2.1.4 Agegroup3 versus Alive: Observed

2.1.5 Agegroup3 versus Alive: Expected

2.1.6 Agegroup3 versus Alive: Chi-square

\(\chi^2 = \sum\left(\frac{(n_{ij} - \mu_{ij})^2}{\mu_{ij}}\right) = \sum\left(\frac{(O - E)^2}{E}\right)=\)

\(\frac{(19 - 151.644)^2}{151.644} + \frac{(141 - 149.116)^2}{149.116} + \frac{(209 - 68.24)^2}{68.24} +\) \(\frac{(521 - 388.356)^2}{388.356} + \frac{(390 - 381.884)^2}{381.884} + \frac{(34 - 174.76)^2}{174.76} =\)

\(116.024 + 0.442 + 290.351 + 45.305 + 0.173 + 113.375 = 565.669\)

2.1.7 Agegroup3 versus Alive: Chi-square

• Degrees of freedom = \((I - 1) \times (J - 1) = (3 - 1) \times (2 - 1) = 2\)
  • \(\chi^2_{critical}(2) = 5.99\)
  • \(565.669 > 5.99\)
  • Reject \(H_0\) that AgeGroup3 and Alive are independent
• But then what?
  • What is different?
  • Similar to ANOVA
    • With 3 groups (levels), which one(s) are different from each other?

2.2.1 Partitioned chi-square

• Chi-square statistics can be split up (partitioned):
  • Alive by (18 to 40 vs 65+)
  • Alive by ((18 to 40 and 41 to 64) vs 65+)
  • These are two independent tests
• Chi-square for all independent tests add up to chi-square for complete table
  • Kind of
• Degrees of freedom also add up

2.2.2 Orthogonal tests

• Orthogonal partitioning of a contingency table is similar to coding orthogonal contrasts for ANOVA

2.2.3 Agegroup3 versus Alive: Partition 1

    Pearson's Chi-squared test

data:  Age3_Alive[c(1, 2), ]
X-squared = 111.79, df = 1, p-value < 0.00000000000000022

2.2.4 Agegroup3 versus Alive: Partition 2

    Pearson's Chi-squared test

data:  Age_Alive
X-squared = 495.33, df = 1, p-value < 0.00000000000000022

2.2.5 Partitioned chi-square

• Overall: \(\chi^2(2) = 565.67\)
• Partition 1: \(\chi^2(1) = 111.79\)
• Partition 2: \(\chi^2(1) = 495.33\)
• \(111.79 + 495.33 = 607.12 \approx 565.67\)
  • For the \(\chi^2\) statistic, the sum will be approximate
    • Closer for larger samples and larger tables
  • A slightly different statistic, \(G^2\), will always sum perfectly
    • \(G^2\) can also be partitioned in the same

2.2.6 \(G^2\) statistic

• \(G^2=2\Sigma\left(n_{ij}\times ln\left(\dfrac{n_{ij}}{\mu_{ij}}\right)\right)\)
  • Also called “likelihood ratio test statistic”
  • Compare to \(\chi^2\) distribution with \((I - 1) \times (J - 1)\) df

2.2.7 \(G^2\) statistic

• Overall \(3 \times 2\) table

    Log likelihood ratio (G-test) test of independence without correction

data:  Age3_Alive
G = 584.41, X-squared df = 2, p-value < 0.00000000000000022

• Just 18 to 40 vs 41 to 64

    Log likelihood ratio (G-test) test of independence without correction

data:  Age3_Alive[c(1, 2), ]
G = 124.02, X-squared df = 1, p-value < 0.00000000000000022

• Combined (18 to 40 and 41 to 64) vs 65+

    Log likelihood ratio (G-test) test of independence without correction

data:  Age_Alive
G = 460.39, X-squared df = 1, p-value < 0.00000000000000022

2.3.1 Residuals

• Residuals exist for \(\chi^2\) just like linear regression

• Raw residual = observed - expected = \(n_{ij} - \hat{\mu}_{ij}\)

• Standardized residual divides by std error of raw residuals

  • \(\frac{n_{ij} - \hat{\mu}_{ij}}{\sqrt{\hat{\mu}_{ij}(1-p_{i+})(1-p_{+j})}}\)
  • where \(\sqrt{\hat{\mu}_{ij}(1-p_{i+})(1-p_{+j})}\) is std error of raw residuals under \(H_0\)
  • and \(p_{i+} = n_{i+}/n\) and \(p_{+j} = n_{+j}/n\)

2.3.2 Observed and expected frequencies

2.3.3 Residuals

2.3.4 Standardized residuals

• Under \(H_0\), variables are independent
  • Observed cell frequencies = expected cell frequencies
    • Residuals tell you how much each cell deviates from this
    • Large standardized residual = cell shows lack of fit from \(H_0\)
• Standardized \(\approx\) normal distribution
  • Expect about 5% of residuals to be greater than \(\pm 2\)
    • Look at standardized residual greater than \(\pm 2\)
    • In small tables, this is way off

2.3.5 Standardized residuals

3.1.1 Adding a 3rd variable

• Control for a potentially confounding third variable, like smoking
  • Relationship between AgeGroup (\(X\)) and Alive (\(Y\))
    • What if smokers have one relationship between \(X\) and \(Y\)
    • But non-smokers have a different relationship between \(X\) and \(Y\)?
  • From last time: Smoker and Alive had an unexpected pattern
    • Smokers were less likely to die than non-smokers?
    • What if smokers are younger than non-smokers and that’s what’s really going on?

3.1.2 Adding a third variable: \(Z\) = Smoker

3.1.3 3 variables = 3-way = 3D

• 2 variables = 2-way or 2D table
  • 3 variables = 3-way or 3D table
• Two ways to look at a 3D table
  • Partial table (a.k.a. conditional table)
  • Marginal table

3.1.4 Partial tables

, , Smoker = No

          Alive
AgeGroup   Dead Alive  Sum
  18 to 64   65   474  539
  65+       165    28  193
  Sum       230   502  732

, , Smoker = Yes

          Alive
AgeGroup   Dead Alive  Sum
  18 to 64   95   437  532
  65+        44     6   50
  Sum       139   443  582

, , Smoker = Sum

          Alive
AgeGroup   Dead Alive  Sum
  18 to 64  160   911 1071
  65+       209    34  243
  Sum       369   945 1314

3.1.5 Marginal table

          Alive
AgeGroup   Dead Alive  Sum
  18 to 64  160   911 1071
  65+       209    34  243
  Sum       369   945 1314

3.1.6 Alive vs AgeGroup: Conditional on Smoker

, , Smoker = No

          Alive
AgeGroup   Dead Alive  Sum
  18 to 64   65   474  539
  65+       165    28  193
  Sum       230   502  732

, , Smoker = Yes

          Alive
AgeGroup   Dead Alive  Sum
  18 to 64   95   437  532
  65+        44     6   50
  Sum       139   443  582

, , Smoker = Sum

          Alive
AgeGroup   Dead Alive  Sum
  18 to 64  160   911 1071
  65+       209    34  243
  Sum       369   945 1314

# A tibble: 4 × 6
# Groups:   AgeGroup, Smoker [4]
  AgeGroup Smoker Alive count  prop totaln
  <fct>    <fct>  <int> <int> <dbl>  <int>
1 18-64    No         1   474 0.879    539
2 18-64    Yes        1   437 0.821    532
3 65+      No         1    28 0.145    193
4 65+      Yes        1     6 0.12      50

3.1.7 Alive vs Smoker: Conditional on AgeGroup

, , AgeGroup = 18-64

      Alive
Smoker Dead Alive  Sum
   No    65   474  539
   Yes   95   437  532
   Sum  160   911 1071

, , AgeGroup = 65+

      Alive
Smoker Dead Alive  Sum
   No   165    28  193
   Yes   44     6   50
   Sum  209    34  243

, , AgeGroup = Sum

      Alive
Smoker Dead Alive  Sum
   No   230   502  732
   Yes  139   443  582
   Sum  369   945 1314

# A tibble: 4 × 5
# Groups:   Smoker, AgeGroup [4]
  Smoker AgeGroup Alive count  prop
  <fct>  <fct>    <int> <int> <dbl>
1 No     18-64        1   474 0.879
2 No     65+          1    28 0.145
3 Yes    18-64        1   437 0.821
4 Yes    65+          1     6 0.12 

3.1.8 Alive vs AgeGroup: Ignoring (marginal on) Smoker

          Alive
AgeGroup   Dead Alive  Sum
  18 to 64  160   911 1071
  65+       209    34  243
  Sum       369   945 1314

# A tibble: 2 × 4
# Groups:   AgeGroup [2]
  AgeGroup Alive count  prop
  <fct>    <int> <int> <dbl>
1 18-64        1   911 0.851
2 65+          1    34 0.140

3.1.9 Alive vs Smoker: Ignoring (marginal on) AgeGroup

      Alive
Smoker Dead Alive  Sum
   No   230   502  732
   Yes  139   443  582
   Sum  369   945 1314

# A tibble: 2 × 5
# Groups:   Smoker [2]
  Smoker Alive count  prop totaln
  <fct>  <int> <int> <dbl>  <int>
1 No         1   502 0.686    732
2 Yes        1   443 0.761    582

3.1.10 Plot: Proportion Alive by AgeGroup, by Smoker

• Conditional effect of AgeGroup, conditional on Smoker…

3.1.11 Plot: Proportion Alive by AgeGroup, by Smoker

• …plus marginal effect, ignoring Smoker

3.1.12 Plot: Proportion Alive by Smoker, by AgeGroup

• Conditional effect of Smoker, conditional on AgeGroup…

3.1.13 Plot: Proportion Alive by Smoker, by AgeGroup

• …plus marginal effect, ignoring AgeGroup

3.2.1 Conditional vs marginal effects

• Conditional effects take \(Z\) into account
  • Smokers less likely to be alive than non-smokers among young people
  • Smokers less likely to be alive than non-smokers among old people
• Marginal effects ignore \(Z\)
  • Smokers more likely to be alive than non-smokers

3.2.2 Simpson’s paradox

• Opposite direction marginal vs conditional effects
• Not unique to contingency tables
  • Can happen any time you ignore a confounder
  • Change in direction of effect after adding covariate
• Related to
  • Lord’s paradox: ANCOVA versus difference scores
  • Suppression effects: Change relationship by adding covariate
  • Ecological fallacy: Opposite direction effects at higher vs lower level

3.2.3 Why does it happen?

• Relationship between the confounder and other variables besides \(Y\)
  • In this case, the relationship between AgeGroup and Smoker

          Smoker
AgeGroup     No  Yes  Sum
  18 to 64  539  532 1071
  65+       193   50  243
  Sum       732  582 1314

• Younger people are \(\frac{532/539}{50/193} = \frac{0.987}{0.259} = 3.81\) times more likely to be smokers than older people
  • \(1071/243 = 4.41\) times more young people than older people
  • A lot of young people, who are more likely to smoke and less likely to die

3.2.4 Plot: Simpson’s paradox

3.2.5 Plot: Total \(n\) for each proportion

3.2.6 Simpson’s paradox

• Not really a paradox
  • Just a different way of looking at the information
• When you have 3 variables
  • Focus is \(XY\) relationship
  • But also pay attention to the \(ZY\) and \(XZ\) relationships

3.3.1 Conditional and marginal effects

• Partial tables
  • Conditional associations odds ratios, chi-square
  • Association between \(X\) and \(Y\), at a given value of \(Z\)
• Marginal tables
  • Marginal associations, odds ratios, chi-square
  • Association between \(X\) and \(Y\), ignoring \(Z\)

3.3.2 Partial tables

, , AgeGroup = 18-64

      Alive
Smoker Dead Alive  Sum
   No    65   474  539
   Yes   95   437  532
   Sum  160   911 1071

, , AgeGroup = 65+

      Alive
Smoker Dead Alive  Sum
   No   165    28  193
   Yes   44     6   50
   Sum  209    34  243

, , AgeGroup = Sum

      Alive
Smoker Dead Alive  Sum
   No   230   502  732
   Yes  139   443  582
   Sum  369   945 1314

3.3.3 Conditional odds ratios: Young people

• \(\hat{\theta} = \frac{474/65}{437/95} = \frac{7.292}{4.6} = 1.585\)
• Odds of non-smoker being alive = \(7.292\)
  • A non-smoker is \(7.292\) times more likely to be alive than dead
• Odds of smoker being alive = \(4.6\)
  • A smoker is \(4.6\) times more likely to be alive than dead
• Odds ratio = \(1.585\): Odds of a non-smoker being alive is \(1.585\) times the odds of an smoker being alive
  • Non-smokers are more likely to be alive than smokers

3.3.4 Conditional odds ratios: Older people

• \(\hat{\theta} = \frac{28/165}{6/44} = \frac{0.170}{0.136} = 1.248\)
• Odds of non-smoker being alive = \(0.170\)
  • A non-smoker is \(0.170\) times more likely to be alive than dead
• Odds of smoker being alive = \(0.136\)
  • A smoker is \(0.136\) times more likley to be alive than dead
• Odds ratio = \(1.248\): Odds of a non-smoker being alive is \(1.248\) times the odds of an smoker being alive
  • Non-smokers are more likely to be alive than smokers

3.3.5 Marginal table

      Alive
Smoker Dead Alive  Sum
   No   230   502  732
   Yes  139   443  582
   Sum  369   945 1314

3.3.6 Marginal odds ratio

• \(\hat{\theta} = \frac{502/230}{443/139} = \frac{2.183}{3.187} = 0.685\)
• Odds of non-smoker being alive = \(2.183\)
  • A non-smoker is \(2.183\) times more likely to be alive than dead
• Odds of smoker being alive = \(3.187\)
  • A smoker is \(3.187\) times more likley to be alive than dead
• Odds ratio = \(0.685\): Odds of a non-smoker being alive is \(0.685\) times the odds of an smoker being alive
  • Non-smokers are less likely to be alive than smokers

3.3.7 Conditional and marginal effects

• Neither tell the whole story usually
  • Look at both
• We looked at odds ratios
  • Also consider conditional and marginal difference in proportion and relative risk, if applicable

3.4.1 Conditional vs marginal independence

• Variables can be
  • Conditionally independent
  • Marginally independent
  • Both
  • Neither
• Depends on \(XY\) relationship, as well as \(ZY\) and \(XZ\) relationships

3.4.3 … but not marginal independence

• Combine both clinics into a single large sample

• Odds of success for treatment A is twice odds of success for treatment B
• OR = \(\frac{20/20}{20/40} = \frac{1}{0.5} = 2\)

3.4.4 What to do?

• Look at both partial and marginal tables

• Unless \(XY\) relationship is same at every level of \(Z\), partial and marginal effects will tell different stories

• Very similar to covariates in ANOVA or regression

  • Main effect for \(XY\) relationship doesn’t tell you the complete story if
    • Covariate is related to the outcome
    • Covariate and predictor are related
    • Interaction of covariate and predictor

4.1.1 Summary of this week

• Extend to more levels and/or more variables
  • Additional complexity with
    • Partitioning effects
    • Partial vs marginal effects

4.1.2 Summary of this section

• Contingency tables
  • \(2 \times 2\) and larger
• Measures of relationship
  • Difference in proportion
  • Relative risk
  • Odds ratio
  • Chi-square tests
• Extend these to larger tables